C3 chapter 3 - exponentials and logarithms

This chapter is about the exponential function and the natural logarithm.

Sections

C3 section 3.1

Before starting this section you should:

• understand the use and laws of indices from C1 chapter 1 - manipulating algebra and surds
• be able to sketch curves from tables of values as we did in C1 chapter 2 - quadratic functions and
• have a good intuitive understanding of gradient as a function that rests on your understanding of derivatives from C1 chapter 7 - differentiation ,

Exponential functions are of the form

$y = a^x \ for \ a \in \mathbb{R} \ , \ a > 0$

We know that graphs of this form must pass through (0,1) since

$a^0 \equiv 1 \ for \ all \ a \in \mathbb{R} , a \not= 0$

Now there are three cases to consider for this sort of function:

1.	$y = a^x \ for \ a \in \mathbb{R} \ , \ a = 1$
2.	$y = a^x \ for \ a \in \mathbb{R} \ , \ a > 1$
3.	$y = a^x \ for \ a \in \mathbb{R} \ , \ 0 < a < 1$

You can experiment with these graphs in the interactive versions below:

and

Examples:

Now it's a reasonable question to ask: "what is the gradient of each of these graphs as x changes?"
That is to say

We will learn in C4 that the derivative of an exponential function is given by:

But for now, since you haven't met "ln" let's explore an interactive graph. The red line is the original exponential function and the blue line shows the gradient of the red line at that value of x:

Play with the slider on the graph to make the blue and red lines coincide. This means that for this value of a, the gradient of the graph is exactly equal to the value of the graph. Roughly what value of a achieves this?
Here are some of my efforts:

the gradient is too small	the gradient is just a bit too small	the gradient is just about right	the gradient is just a bit too big	the gradient is too big

C3 section 3.2

Before starting this section you need to know the transformation effects on graphs of changing y = f(x) into:

• y = f(x) + c
• y = af(x)
• y = -f(x)
• y = f(x + c)
• y = f(ax)
• y = f(-x)

You should also know that the graphs of y = f(x) and its inverse function y = f -1(x) are reflections in y = x.
This section is about the characteristic shapes of exponential graphs

C3 Exercise 3A

• C3 Exercise 3A worked solutions:
  

  C3 Exercise 3A (using the exponential function).jnt
  

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• Questions similar to this exercise:

C3 section 3.3

You need to understand range and domain from C3 chapter 2.

The inverse function of an exponential function is a logarithm to the same base.
Given

The inverse function of the exponential function is the natural logarithm.

You need to remember the laws for logs:
$\log_x a + \log_x b = \log_x ab$
and in particular this works for the natural logarithm
$\ln a + \ln b = \ln ab$

and the corresponding laws for division:
$\log_x a - \log_x b = \log_x \frac{a}{b}$
and for the natural logarithm
$\ln a - \ln b = \ln \frac{a}{b}$

C3 Exercise 3B

• C3 Exercise 3B worked solutions:
  

  C3 Exercise 3B (ln and e^x graphs) B.jnt
  

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C3 Exercise 3C

• C3 Exercise 3C worked solutions to part of the exercise:
  

  C3 Exercise 3C (exponential & log graphs).jnt
  

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